We propose a theory to explain random behavior for the digits in the expansions of fundamental mathematical constants. At the core of our theory is a general hypothesis concerning the distribution of the iterates generated by a certain dynamical map. On this main hypothesis, one obtains proofs of normality (digit randomness in a specific technical sense) for a collection of celebrated constants, including pi, log 2, and others. A connections is drawn between the dynamical model and alternative statistical pictures such as a theory of cascaded pseudorandom number generators. A connection is also established with the theory of irrational and transcendental numbers, leading tot he possibility of normality proofs for number-theoretically motivated entities such as the Erdos-Borwein and Euler constants.